Metrics and [formula omitted]-Equalities

Abstract

The relationship between metrics and T-equalities is investigated; the latter are a special case of T-equivalences, a natural generalization of the classical concept of an equivalence relation. It is shown that in the construction of metrics from T-equalities triangular norms with an additive generator play a key role. Conversely, in the construction of T-equalities from metrics this role is played by triangular norms with a continuous additive generator or, equivalently, by continuous Archimedean triangular norms. These results are then applied to the biresidual operator ET of a triangular norm T. It is shown that ET is a T-equality on [0, 1] if and only if T is left-continuous. Furthermore, it is shown that to any left-continuous triangular norm T there correspond two particular T-equalities on F(X), the class of fuzzy sets in a given universe X; one of these T-equalities is obtained from the biresidual operator ETT by means of a natural extension procedure. These T-equalities then give rise to interesting metrics on F(X).